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Planar inviscid flows in a channel of finite length: washout, trapping and self-oscillations of vorticity

Published online by Cambridge University Press:  09 July 2010

V. N. GOVORUKHIN ,
A. B. MORGULIS  and
V. A. VLADIMIROV
V. N. GOVORUKHIN
Affiliation:
Department of Mathematics and Mechanics, Southern Federal University, Melchakova Street 8a, 344090 Rostov-on-Don, Russia
A. B. MORGULIS
Affiliation:
Department of Mathematics and Mechanics, Southern Federal University, Melchakova Street 8a, 344090 Rostov-on-Don, Russia Southern Institute of Mathematics, Russian Academy of Sciences, Marcus Street 22, 362027 Vladikavkaz, Russia
V. A. VLADIMIROV*
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK
*
†Email address for correspondence: vv500@york.ac.uk
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Abstract

The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

JFM classification

Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 420 - 472
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

(This paper is dedicated to the memory of Victor Yudovich)

References

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